Mularkey was hired by a GM, Gene Smith, who is long gone. Caldwell needs to have the freedom to remake the organization in his image. He always would want his "own" coach hire, so it's better to make a clean break with Mularkey now.
We've seen new GMs hang on to inherited coaches, but it rarely lasts for long. Think Ted Thompson and Mike Sherman together in Green Bay. Think Mike Holmgren wasting a year with Eric Mangini in Cleveland. Think Lovie Smith being fired after a 10-6 season because Chicago Bears GM Phil Emery wanted his own guy.
The Jaguars are a mess. Whoever takes over will inherit one of the most talent-poor rosters in the NFL. Blaine Gabbert will have his third head coach in as many seasons; he'll have to battle for playing time. Chad Henne is under contract next year. The team's long-term answer at quarterback is not on the roster. Mularkey didn't show anything to make his bosses believe he could turn around a lousy squad.
Mularkey probably knows he's not going to get another bite at the NFL head-coaching apple. He walked away from a head-coaching job in Buffalo in odd circumstances after two seasons, then lasted only one year in Jacksonville. He should land on his feet as a coordinator somewhere, and he is regarded as a very good man in the business.